Linear Programming Skill 45

Objective HSA-REI.12(2): Solve

Linear programming is method of finding a maximum or minimum value of a function that satisfies a given set of conditions called constraints. A constraint is one of the inequalities in a linear programming problem. The solution to the set of constraints can be graphed as a feasible region.

Example: Linear Programming Lauren wants to paint no more than 70 plates for the art show. It costs her at least $50 plus $2 per item to produce red plates and $3 per item to produce gold plates. She wants to spend no more than $215. Write and graph a system of inequalities that can be used to determine the number of each plate that Lauren can make. x represent the number of red plates y represent the number of gold plates.

Example: Linear Programming y  0 The system of inequalities is x + y ≤ 70 50 + 2x + 3y ≤ 215

(x, y) 10x+15y P($) (0,0) 10(0)+15(0) (0,55) 10(0)+15(55) 850 (45,25) Suppose Lauren sells the red plates for $10 each and the gold plates for $15 each. How many of each type of plate should she paint to maximize her profit? What is her maximum profit. (x, y) 10x+15y P($) (0,0) 10(0)+15(0) The maximum value is at the vertex (0, 55). (0,55) 10(0)+15(55) 850 (45,25) 10(45)+15(25) 825 (70,0) 10(70)+15(0) 700 Lauren should paint 55 gold plates and 0 red plates to maximize the amount of profit, which is $850.

Example; Linear Programming Leyla is selling hot dogs and spicy sausages at the fair. She has only 40 buns, so she can sell no more than a total of 40 hot dogs and spicy sausages. Each hot dog sells for $2, and each sausage sells for $2.50. Leyla needs at least $90 in sales to meet her goal. Write and graph a system of inequalities that models this situation. d represent the number of hot dogs s represent the number of sausages

Do It Yourself: Example; Application The system of inequalities is d + s ≤ 40 2d + 2.5s ≥ 90

(d, b) 2d + 2.5b P($) (0,36) 2(0)+2.5(0) (0,40) 2(0)+2.5(40) 100 How many of hotdogs (d) and sausages (s) must she sell to maximize her profit? What is her maximum profit. (d, b) 2d + 2.5b P($) (0,36) 2(0)+2.5(0) The maximum value is at the vertex (0, 40). (0,40) 2(0)+2.5(40) 100 (20,20) 2(20)+2.5(20) 90 Leyla should sell 0 hotdogs and 40 sausages to maximize the amount of profit, which is $100.

#45: Linear Programming Questions Summarize Notes Homework Google Classroom Quiz